Projectivization

The projectivization of a vector space over a field , sometimes written , is the set of equivalence classes of nonzero vectors under the relation , where is nonzero. Equivalently, its points are the one-dimensional subspaces of . For example, complex projective space has homogeneous coordinates , with not all .

If is a nonzero finite-dimensional real vector space or complex vector space of dimension , its projectivization is a manifold of dimension over the same field. Thus the real dimension of complex projective space is . These projectivizations are covered by affine coordinate charts,